Also you will get a different amount of overtones out of a drum depending on how you hit it. In the centre = less overtones. Off centre = more overtones. I usually hit my snare a little off centre because I like the way it sounds like that.

I think you may be confusing ring with overtones. Pocket's description is correct, overtones are part of any instrument's sound, and all instruments produce overtones. They're an integral part of an instrument's sound.